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Theorem disjabrexf 31501
Description: Rewriting a disjoint collection into a partition of its image set. (Contributed by Thierry Arnoux, 30-Dec-2016.) (Revised by Thierry Arnoux, 9-Mar-2017.)
Hypothesis
Ref Expression
disjabrexf.1 𝑥𝐴
Assertion
Ref Expression
disjabrexf (Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}𝑦)
Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝐴,𝑧   𝑦,𝐵,𝑧
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem disjabrexf
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfdisj1 5084 . . . 4 𝑥Disj 𝑥𝐴 𝐵
2 nfcv 2907 . . . . 5 𝑥𝑦
3 disjabrexf.1 . . . . . . . . . . 11 𝑥𝐴
43nfcri 2894 . . . . . . . . . 10 𝑥 𝑖𝐴
5 nfcsb1v 3880 . . . . . . . . . . 11 𝑥𝑖 / 𝑥𝐵
65nfcri 2894 . . . . . . . . . 10 𝑥 𝑗𝑖 / 𝑥𝐵
74, 6nfan 1902 . . . . . . . . 9 𝑥(𝑖𝐴𝑗𝑖 / 𝑥𝐵)
87nfab 2913 . . . . . . . 8 𝑥{𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)}
98nfuni 4872 . . . . . . 7 𝑥 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)}
109nfcsb1 3879 . . . . . 6 𝑥 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵
1110nfeq1 2922 . . . . 5 𝑥 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦
122, 11nfralw 3294 . . . 4 𝑥𝑗𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦
13 eqeq2 2748 . . . . 5 (𝑦 = 𝐵 → ( {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝐵))
1413raleqbi1dv 3307 . . . 4 (𝑦 = 𝐵 → (∀𝑗𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦 ↔ ∀𝑗𝐵 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝐵))
15 vex 3449 . . . . 5 𝑦 ∈ V
1615a1i 11 . . . 4 (Disj 𝑥𝐴 𝐵𝑦 ∈ V)
17 simplll 773 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → Disj 𝑥𝐴 𝐵)
18 simpllr 774 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → 𝑥𝐴)
19 simprl 769 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → 𝑖𝐴)
20 simplr 767 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → 𝑗𝐵)
21 simprr 771 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → 𝑗𝑖 / 𝑥𝐵)
22 csbeq1a 3869 . . . . . . . . . . . . . 14 (𝑥 = 𝑖𝐵 = 𝑖 / 𝑥𝐵)
233, 5, 22disjif2 31499 . . . . . . . . . . . . 13 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑖𝐴) ∧ (𝑗𝐵𝑗𝑖 / 𝑥𝐵)) → 𝑥 = 𝑖)
2417, 18, 19, 20, 21, 23syl122anc 1379 . . . . . . . . . . . 12 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → 𝑥 = 𝑖)
25 simpr 485 . . . . . . . . . . . . . 14 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → 𝑥 = 𝑖)
26 simpllr 774 . . . . . . . . . . . . . 14 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → 𝑥𝐴)
2725, 26eqeltrrd 2839 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → 𝑖𝐴)
28 simplr 767 . . . . . . . . . . . . . 14 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → 𝑗𝐵)
2922eleq2d 2823 . . . . . . . . . . . . . . 15 (𝑥 = 𝑖 → (𝑗𝐵𝑗𝑖 / 𝑥𝐵))
3025, 29syl 17 . . . . . . . . . . . . . 14 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → (𝑗𝐵𝑗𝑖 / 𝑥𝐵))
3128, 30mpbid 231 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → 𝑗𝑖 / 𝑥𝐵)
3227, 31jca 512 . . . . . . . . . . . 12 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → (𝑖𝐴𝑗𝑖 / 𝑥𝐵))
3324, 32impbida 799 . . . . . . . . . . 11 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → ((𝑖𝐴𝑗𝑖 / 𝑥𝐵) ↔ 𝑥 = 𝑖))
34 equcom 2021 . . . . . . . . . . 11 (𝑥 = 𝑖𝑖 = 𝑥)
3533, 34bitrdi 286 . . . . . . . . . 10 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → ((𝑖𝐴𝑗𝑖 / 𝑥𝐵) ↔ 𝑖 = 𝑥))
3635abbidv 2805 . . . . . . . . 9 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = {𝑖𝑖 = 𝑥})
37 df-sn 4587 . . . . . . . . 9 {𝑥} = {𝑖𝑖 = 𝑥}
3836, 37eqtr4di 2794 . . . . . . . 8 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = {𝑥})
3938unieqd 4879 . . . . . . 7 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = {𝑥})
40 unisnv 4888 . . . . . . 7 {𝑥} = 𝑥
4139, 40eqtrdi 2792 . . . . . 6 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = 𝑥)
42 csbeq1 3858 . . . . . . 7 ( {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = 𝑥 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑥 / 𝑥𝐵)
43 csbid 3868 . . . . . . 7 𝑥 / 𝑥𝐵 = 𝐵
4442, 43eqtrdi 2792 . . . . . 6 ( {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = 𝑥 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝐵)
4541, 44syl 17 . . . . 5 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝐵)
4645ralrimiva 3143 . . . 4 ((Disj 𝑥𝐴 𝐵𝑥𝐴) → ∀𝑗𝐵 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝐵)
471, 12, 14, 16, 46elabreximd 31436 . . 3 ((Disj 𝑥𝐴 𝐵𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}) → ∀𝑗𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦)
4847ralrimiva 3143 . 2 (Disj 𝑥𝐴 𝐵 → ∀𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}∀𝑗𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦)
49 invdisj 5089 . 2 (∀𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}∀𝑗𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}𝑦)
5048, 49syl 17 1 (Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  {cab 2713  wnfc 2887  wral 3064  wrex 3073  Vcvv 3445  csb 3855  {csn 4586   cuni 4865  Disj wdisj 5070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-sn 4587  df-pr 4589  df-uni 4866  df-disj 5071
This theorem is referenced by:  measvunilem  32811
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